thermo functions

Section 4.3

Solid Mechanics Part III Kelly 414

4.3 Thermodynamic Functions Four important and useful thermodynamic functions will be considered in this section (two of them have been encountered in the previous sections). These are the internal energy U, the enthalpy H, the Helmholtz free energy (or simply the free energy) and the Gibbs free energy (or simply the Gibbs function) G. These functions will be defined and examined below for both reversible and irreversible processes. 4.3.1 Reversible Processes Consider first a reversible process. The Internal Energy The internal energy is

dSpdVQWdU

(4.3.1)

the second line being valid for quasi-static processes. The properties of a pure compressible substance include SV ,, and p. From 4.3.1, it is natural to take V and S as the state variables:

dSSUdV

VUdU

VS

(4.3.2)

so that

VS SU

VUp

, (4.3.3)

Thus ),( SVU contains all the thermodynamic information about the system; given V and S one has an expression for U and can evaluate p and through differentiation. U is a thermodynamic potential, meaning that it provides information through a differentiation. V and S are said to be the canonical (natural) state variables for U. By contrast, expressing the internal energy as a function of the volume and temperature, for example,

),( VUU , is not so useful, since this cannot provide all the necessary information regarding the state of the material. A new state function will be introduced below which has V and as canonical state variables. Similarly, the equation of state ),( pV does not contain all the thermodynamic information. For example, there is no information about U or S, and this equation of state must be supplemented by another, just as the ideal gas law is supplemented by the caloric equation of state )(UU .

Section 4.3


Returning to the internal energy function, and taking the differential relations between

,p and U, Eqns. 4.3.3, and differentiating them again, and using the fact that VSUSVU // 22 , one arrives at the Maxwell relation,

SV VSp

(4.3.4)

The Helmholtz Free Energy Define the (Helmholtz) free energy function through

SU (4.3.5) One has

SdpdVdSQdUdSdSdUd

, (4.3.6)

the second line being valid for reversible processes. Now V and have emerged as the natural state variables. Writing ),( V ,

ddVVd V

(4.3.7)

so that

V

SV

p

, (4.3.8)

The Enthalpy and Gibbs Free Energy The enthalpy is defined by Eqn. 4.1.18,

pVUH (4.3.9) To determine the canonical state variables, evaluate the increment:

dH dU pdV Vdp

W Q pdV Vdp (4.3.10)

and so

VdpdSdH (4.3.11)

Section 4.3


and the natural variables are p and S . Finally, the Gibbs free energy function is defined by

pVSUG (4.3.12) and the canonical state variables are p and . The definitions, canonical state variables and Maxwell relations for all four functions are summarised in Table 4.3.1 below. Thermo-dynamic potential

Symbol and

appropriate variables

Definition Differential relationship Maxwell relation

Internal energy ),( VSU dSpdVdU

VS

SV

Sp

V

VUp

SU

,

Enthalpy ),( pSH pVUH dSVdpdH pS

Sp

SV

p

pHV

SH

,

Helmholtz free

energy ),( V SU SdpdVd

V

Vp

VS

VpS

,

Gibbs free energy ),( pG pVSUG SdVdpdG

p

p

VpS

pGVGS

,

Table 4.3.1: Thermodynamic Potential Functions and Maxwell relations Mechanical variables: whereas the internal energy and the Helmholtz free energy are functions of a kinematic variable (V), the enthalpy and the Gibbs function are functions of a force variable (p). Thermal variables: whereas the internal energy and the enthalpy are functions of the entropy, the Helmholtz and Gibbs free energy functions are functions of the temperature. If one is analyzing a process with, for example, constant temperature, it makes sense to use either the Helmholtz or Gibbs free energy functions, so that there is only one variable to consider. Note that the temperature is an observable property and can be controlled to some extent. Values for the entropy, on the other hand, cannot be assigned arbitrary values in experiments. For this reason a description in terms of the free energy, for example, is often more useful than a description in terms of the internal energy.

Section 4.3


4.3.2 Irreversible Processes Consider now an irreversible process. The Internal Energy One has )(iSdSdUW and, with the internal energy again a function of the entropy and volume,

)(i

SV

SdVVUdS

SUW

(4.3.13)

Consider the case of pure heating 0)( iSdVW , so

VSU

(4.3.14)

as in the reversible case. This relation between properties is of course valid for any process, not necessarily a pure heating one. Thus

)(i

S

SdVVUW

(4.3.15)

Express the work in the form

dVAdVAWdWW

dq

dq

)()(

)()(

(4.3.16)

such that the quasi-conservative force )(qA is that associated with the work )(qW which is recoverable, whilst the dissipative force dA produces the work )(dW which is dissipated, i.e. associated with irreversibilities. From 4.3.15,

S

q

VUA

)( (4.3.17)

and the dissipative work )(dW is

0)()()( idd SdVAW (4.3.18) The name quasi-conservative force for the )(qA (here, actually a force per area) is in recognition that the internal energy plays the role of a potential in 4.3.17, but it is also a function of the entropy. It can be seen from Eqn. 4.3.17 that the quasi-conservative force is a state function, and equals p in a fully reversible process.

Section 4.3


In the isentropic case, 0dS , and one has

( )iW dU S (4.3.19) This shows that, in the isentropic case, the internal energy is that part of the work which is recoverable. The Free Energy Directly from SU , with and V the independent variables,

SSU

, VS

VU

V

(4.3.20)

From the pure heating analysis given earlier, Eqn. 4.2.9-10, the term // SU is zero, so

V

S

(4.3.21)

as in the reversible case and

SddVV

d

(4.3.22)

The work can now be written again as Eqn. 4.3.16, but now with the quasi-conservative force given by {Problem 3}

V

A q)( (4.3.23)

The dissipative work is again given by 4.3.18. Also, pA q )( for a reversible process. In the isothermal case, 0d ,

)(iSdW (4.3.24) This shows that, in the isothermal case, the free energy is that part of the work which is recoverable. The quasi-conservative forces for the internal energy and free energy are listed in Table 4.3.2. Note that expressions for quasi-conservative forces are not available in the case of the Enthalpy and Gibbs free energy since they do not permit in their expression increments in volume dV , which are required for expressions of work increment.

Section 4.3


Thermo-dynamic potential Differential relationship Relations

),( VSU )(iSdSpdVdU S

q

V VUA

SU

)(,

SUV ),( )(iSSdpdVd ( ), qv

S AV

Table 4.3.2: Quasi-Conservative Forces for Irreversible Processes, pdVSdVAW iq )()(

4.3.3 The Legendre Transformation The thermodynamic functions can be transformed into one another using a mathematical technique called the Legendre Transformation. The Legendre Transformation is discussed in detail in Part IV, where it plays an important role in Plasticity Theory, and other topics. For the present purposes, note that the Legendre transformation of a function ),( yxf is the function ),( g where

),(),( yxfyxg (4.3.25) and

yf

xf

, ,

gygx , (4.3.26) When only one of the two variables is switched, the transform reads

),(),( yxfxyg (4.3.27) where

xf ,

gx (4.3.28) For example, if one has the function ),( VSU and wants to switch the independent variable from S to , Eqn. 4.3.27 leads one to consider the new function

),(),( VSUSVg (4.3.29) and Eqns. 4.3.28 give

VSU

and

V

gS

(4.3.30)

It can be seen that ),( Vg is the negative of the Helmholtz free energy and the two differential relations in 4.3.30 are contained in Table 4.3.1.

Section 4.3


4.3.4 Problems 1. By considering reversible processes, derive the differential relationships and the

Maxwell relations given in Table 4.3.1 for (a) the enthalpy, (b) the Gibbs free energy 2. Let the two independent variables be V and . Consider the internal energy,

),( VUU . Use the pure heating example considered in 4.2 to show that the quasi-conservative force of Eqn. 4.3.17 can also be expressed as

VS

VUA q)(

3. Show that Eqn. 4.3.22 leads to Eqn. 4.3.23. 4. Use the Legendre Transformation rule to transform the Helmholtz free energy

),( V into a function of the variables and . Derive also the two differential relations analogous to Eqns. 4.3.30. Show that this new function is the negative of the Gibbs energy (use the relation SU ), where p , and that the two differential relations correspond to two of the relations in Table 4.3.1.

5. Use the Legendre Transformation rule to transform the enthalpy ),( pSH into a

function of the variables S and V . Show that this new function is the negative of the internal energy, and that the two differential relations correspond to two of the relations in Table 4.3.1.

thermo functions

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